Re: Response to Brett Barbaro: Pythagorean tuning

Hello, there, and mainly I would like respond to support some of the
points concerning Pythagorean tuning made by Brett Barbaro, and to
expand on a few points concerning stable and _relatively_ concordant
sonorities in some of the main dialects of composed Western European
music during the era 1200-1400 or so.

First of all, in discussing the problems of evaluating a scale and the
vertical intervals it makes possible, I would emphasize a point which
one of Brett's messages makes (at least for me) in discussing
technological questions: the best technologies can hardly substitute
for an understanding of the _style_ of music 1000 years old -- or, in
the case of my focus here, between about 600 and 800 years old.

Secondly, it is a special pleasure on this Tuning List to honor the
germinal contributions of Joseph Yasser in his articles of some 60
years ago on _Medieval Quartal Harmony: A Plea for Restoration_. While
my analysis of medieval polyphony differs significantly in some
respects from Yasser's, his central point cannot too strongly be
affirmed: medieval polyphony based on stable fifths and fourths is a
"harmonic universe" in its own right, and must be judged in its own
terms.

Brett has nicely captured one aspect of this universe: the organizing
of an intervallic spectrum in terms of chains of pure fifths. It may
been seen as a delightful form of serendipity that the same tuning
which yields pure fifths and fourths also happens to yield an
intriguing universe of other intervals, and composers from Perotin
(around 1200) through Machaut (c. 1300-1377) and his successors seek
to explore the vertical potential of _all_ these intervals.

Here it may be well to emphasize that to endorse Pythagorean tuning
for Continental Western European polyphony between 1200 and 1400 is
_not_ to prescribe it for _all_ musics. I would no more recommend
Pythagorean for Palestrina, where thirds and sixths are the principal
concords, than I would recommend 1/4-comma meantone for Perotin, where
Pythagorean is beautifully "in tune" both acoustically and
stylistically. Similarly, it would be rather comically prescriptive to
urge players of 16th-century keyboard music to use a 7:4 minor seventh
because "the normal meantone interval has too high an LCM (infinite
in fact), and so deprives us of one of the most important concords in
music." Nor, I expect, would even the most ardent lover of the
keyboard music of Cabezon or Byrd demand that barbershop quartet
singers perform in correct Renaissance meantone.

Marion has raised the interesting issue of why Ancient Greek theorists
were apparently not concerned with polyphony, and here I would like
only to suggest that the use of the Pythagorean scale does not seem to
me a persuasive explanation. From what I understand, for whatever
reasons, the outlook at least of the Greek practice and theory for
which we have evidence was essentially monophonic. Apparently neither
the tuning of the Pythagoreans with its pure ratios of 6:8:9:12, nor
that of Ptolemy with its intervals of 5:4 and 6:5, nor that of
Aristoxenos with its equal semitones, happened to evolve in a musical
milieu oriented toward polyphony.

However, it also happens that Pythagorean tuning gives superb harmonic
results in the setting of complex Gothic polyphony, and likewise the
tuning of Ptolemy beautifully fits the needs of Renaissance polyphony
(more literally for voices and other non-fixed-pitch instruments, and
with some compromises also on meantone keyboards).

In becoming "acculturated" to any polyphonic music, be it Gothic or
Renaissance, Romantic or contemporary jazz, one must become at home
with the characteristic scale of vertical tension. From Perotin to
Machaut, this scale is a subtle and graduated one. Thus Jacobus of
Liege (c. 1325) describes the major third (81:64) and major second
(9:8) alike as being "neither perfectly concordant nor perfectly
discordant." Both are "compatible," although M3 is somewhat milder or
more blending in itself than M2.

This nuance of Pythagorean tuning, in which the _relatively_ blending
M3 and m3 are made a bit more tense by their rather complex ratios
while the _relatively_ tense M2 (9:8) and m7 (16:9) are made somewhat
more blending by their pure ratios, is vital to what I'm tempted to
call the "jazzy" quality of much 13th-14th century music.

In 13th-century music actually beginning a bit before 1200 with
Perotin and his generation, we have something like the following set
of stable or relatively concordant sonorities available in three-voice
writing. Please note that I identify sonorities in a medieval fashion
by giving the outer interval, lower interval, and upper interval: thus
"8|5-4" means "a sonority with outer octave, lower fifth, and upper
fourth." To clarify what may be an unfamiliar notation, I give also an
example of each sonority, using MIDI-style notation (C4 = middle C) to
show octaves; higher numbers mean higher octaves.

For each sonority, I give Pythagorean ratios both in medieval terms of
string lengths ("wavelength ratios," if you like), and in more modern
terms of frequency ratios. Note how 81:64 and 32:27 are relatively
concordant, and how 9:8, 16:9, and 9:4 become relatively concordant
when combined with two ideally blending intervals of the fifth/fourth
class:

----------------------------------------------------------------------
Richly stable sonorities
----------------------------------------------------------------------
Sonority example string-ratio frequency-ratio
----------------------------------------------------------------------

8|5-4 D3-A3-D4 6:4:3 2:3:4
8|4-5 D3-G3-D4 4:3:2 3:4:6

(Both these sonorities are saturated stable concords, but 8|5-4 is
smoother and more conclusive, while 8|5-4 is relatively stable but
less conclusive.)

----------------------------------------------------------------------
Mildly unstable sonorities
----------------------------------------------------------------------
Sonority example string-ratio frequency-ratio
----------------------------------------------------------------------
5|M3-m3 G3-B3-D4 81:64:54 64:81:96
5|m3-M3 A3-C4-E4 96:81:64 54:64:81

5|4-M2 G3-C4-D4 12:9:8 6:8:9
5|M2-4 G3-A3-D4 9:8:6 8:9:12
m7|4-4 G3-C4-F4 16:12:9 9:12:16
M9|5-5 G3-D4-A4 9:6:4 4:6:9
----------------------------------------------------------------------
----------------------------------------------------------------------

In the 13th century, as Brett eloquently puts it, Pythagorean thirds
and also the somewhat more tense major sixth (27:16, a pure fifth plus
a pure 9:8 major second, often ranked with M2) are not "missed
opportunities" but rather realized opportunities for some of the most
memorable cadences in the history of European music. Indeed, even the
strong discords (m2, M7, A4, d5, and often m6) play an essential role
in polyphonic practice, as the theorist Johannes de Garlandia
(c. 1240?) recognizes.

In the 14th century, this scale of concord shifts somewhat: sixths are
ranked on more of a parity with thirds as relatively concordant, while
fourths above the lowest voice are often regarded as dissonances (a
trend decried by Jacobus, who defends this interval as a full concord,
although more pleasing above the fifth). However, Machaut continues to
use many traditional sonorities such as the major ninth combined with
two euphonious fifths (e.g. F3-C4-G4), while also taking advantage of
combinations involving the Pythagorean major sixth or minor seventh.

From this perspective, Machaut may have considered traditional
cadential sonorities of the 13th century such as M6|5-M2
(e.g. G3-D4-E4) or m7|5-m3 (e.g. E3-B3-D4) not only as somewhat
"compatible" (they contain no strong discords, and Jacobus admits them
as legitimate combinations), but as independently euphonious to a
degree. Both his "modern" penchant for a freer treatment of thirds and
especially sixths, and his "traditional" appreciation for the milder
aspects of M2 and m7, may contribute to this artistic result.

Interestingly, such three-voice sonorities nicely "verticalize" the
common names of the Pythagorean major sixth and minor seventh
themselves: respectively _tonus cum diapente_ (tone plus fifth) and
_semiditonus cum diapente_ (minor third plus fifth). Here are ratios
for the two sonorities just described:

sonority example string-ratio frequency-ratio

M6|5-M2 G3-D4-E4 27:18:16 16:24:27
m7|5-m3 E3-B3-D4 32:24:18 18:27:32

Here it would be beyond of this discussion (already long enough!) to
delve into the cadential treatment of these sonorities, vital in the
13th century and an interesting feature also of Machaut and of some
dialects of 14th-century English music, for example. For one analysis,
please feel welcome to visit:

http://www.medieval.org/emfaq/harmony/13c.html

However, joining Brett Barbaro in a celebration of realized
opportunities in Gothic music, I would add that the m7|5-m3 sonority
has a very pleasant quality to my ear, at once "Bluesy" and sweet,
while the M6|5-M2 is positively "vibrant." At the same time, they are
sonorities of motion, not of rest -- but I could imagine them being
used as very interesting free sonorities in new musics.

Also, in a 14th-century setting, Pythagorean thirds as well as sixths
have a delightful "bounce." Those half-cadences on 5|M3-m3 (to Jacobus
the _quinta fissa_ or "split fifth" sonority, with the middle voice
"splitting" the fifth into two thirds, e.g. G3-B3-D4) are sweet but
indeed inconclusive. The tuning and the music enhance each other, and
as Brett has said, an opportunity is superbly realized.

What implications may this discussion have more generally for tuning
theory? I would suggest that in music, as in the study of natural
languages, theories of "universal grammar" can only be as accurate as
the descriptions of the many specific languages on which they are
premised.

To understand medieval Pythagorean tuning and its possible broader
implications, I would suggest that we begin with the specifics of
13th-14th century vertical style and its interactions with the
tuning. Then we see how these specifics may invite the affirmation or
modification of assumptions about Pythagorean or other just intonation
systems, or about tuning systems in general.

For example, the musically very effective and beautiful use of 81:64
and 32:27 as _relatively concordant_ intervals (both as defined by the
theorists and as realized in the pieces themselves) may indicate that
any theory of "least common multiple" (LCM) must take account of
this artistic possibility. One conclusion that might be that
perceptions of "concord/discord" are largely conditioned by style; in
a 16th-century setting, where the purity of 5:4 and 6:5 thirds is a
cardinal _desideratum_, Zarlino finds that 81:64 would be a
"dissonance."

This does not mean that LCM measures are irrelevant: indeed,
musicians seem often to have a curious habit of seeking low LCM's
for the most choice concords in their "language" (3:2 and 4:3 in the
Gothic; 5:4 and 6:5 in the Renaissance; 7:4 in barbershop; n:m in
whatever n-limit system one happens to prefer at a given moment).

Similarly, "two negatives are equivalent to a positive" is a good rule
for many situations in predicate logic, but does not prevent natural
languages such as Spanish and Middle English from using double or
multiple negatives in the sense of negation.

Further, focusing on the specifics of Gothic style may be only the
beginning of an inquiry into intriguing, if possibly moot, questions
about the fine nuances of tuning this music.

For example, as Marion has ingeniously asked, might performers
realizing the ideal of a Pythagorean intonation on non-fixed-pitch
instruments tend to shade an 81:64 toward a 19:15, or a 729:612 toward
a 10:7? Focusing on the nature of Pythagorean tuning _as a system_ may
make such questions yet more interesting: for example, would an
alteration of this kind (keeping within a range of ~5 cents)
substantially compromise the purity of fifths and fourths? I look
forward to discussing these points in another post.

Now, as some 60 years ago when Yasser wrote, the greatest barrier to
an appreciation of Pythagorean tuning in a Gothic setting would seem
still to be an unfamiliarity with the composed European music of this
period and its distinctive vertical style. As Brett has suggested,
this music is an opportunity which I am very glad not to have missed.

Most respectfully,

Margo Schulter
mschulter@value.net